Stochastic Programming Method Research Articles

A robust mixed-integer binary programming model for operating theater scheduling to the patient and the surgeon under uncertainty in an open-heart Surgery Department

In hospitals, the surgical ward is both a cost and revenue center. In this ward, hospitals face challenges such as increasing demand, limited resources, and rising costs. Consequently, the decisions made have an implications effect on the hospital's performance. Therefore, in this paper a robust mixed-integer binary programming model is proposed with three objectives of maximizing the efficiency of available resources, minimizing the patients waiting time, and minimizing surgery costs that are formulated utilizing the augmented epsilon constraint approach. This model allocates the operating room to the patient and the surgeon and then obtains the required bed capacity inside the downstream units for stand-alone cardiac hospitals. This model includes different preferences for hospital, surgeon, and patient: waiting time, patient cancellations, tardiness, uncertainties in surgery durations, the patient operation start times, the overtime per working day, time windows, SICU beds, planning horizon, and the idle times of the surgeons, operating theater, and working day. The proposed model is solved using robust optimization to deal with stochastic. The proposed model is formulated on the stochastic programming method proposed by Bertsimas and Sim. In the proposed model, a rolling horizon method is used to reschedule the program after cancellation. The computational results illustrate that the rolling horizon method reduces waiting time and increases throughput. The results illustrate that the benefit obtained from the introduced model has improvements in reducing the surgery costs, and patient waiting time, and increasing the efficiency of available resources. This study has been performed at Shahid Rajaei Heart Hospital in Iran.

Decomposition methods for multi-horizon stochastic programming

Multi-horizon stochastic programming includes short-term and long-term uncertainty in investment planning problems more efficiently than traditional multi-stage stochastic programming. In this paper, we exploit the block separable structure of multi-horizon stochastic linear programming, and establish that it can be decomposed by Benders decomposition and Lagrangean decomposition. In addition, we propose parallel Lagrangean decomposition with primal reduction that, (1) solves the scenario subproblems in parallel, (2) reduces the primal problem by keeping one copy for each scenario group at each stage, and (3) solves the reduced primal problem in parallel. We apply the parallel Lagrangean decomposition with primal reduction, Lagrangean decomposition and Benders decomposition to solve a stochastic energy system investment planning problem. The computational results show that: (a) the Lagrangean type decomposition algorithms have better convergence at the first iterations to Benders decomposition, and (b) parallel Lagrangean decomposition with primal reduction is very efficient for solving multi-horizon stochastic programming problems. Based on the computational results, the choice of algorithms for multi-horizon stochastic programming is discussed.

Sample intelligence-based progressive hedging algorithms for the stochastic capacitated reliable facility location problem

Selecting facility locations requires significant investment to anticipate and prepare for disruptive events like earthquakes, floods, or labor strikes. In practice, location choices account for facility capacities, which often cannot change during disruptions. When a facility fails, demand transfers to others only if spare capacity exists. Thus, capacitated reliable facility location problems (CRFLP) under uncertainty are more complex than uncapacitated versions. To manage uncertainty and decide effectively, stochastic programming (SP) methods are often employed. Two commonly used SP methods are approximation methods, i.e., Sample Average Approximation (SAA), and decomposition methods, i.e., Progressive Hedging Algorithm (PHA). SAA needs large sample sizes for performance guarantee and turn into computationally intractable. On the other hand, PHA, as an exact method for convex problems, suffers from the need to iteratively solve numerous sub-problems which are computationally costly. In this paper, we developed two novel algorithms integrating SAA and PHA for solving the CRFLP under uncertainty. The developed methods are innovative in that they blend the complementary aspects of PHA and SAA in terms of exactness and computational efficiency, respectively. Further, the developed methods are practical in that they allow the specialist to adjust the tradeoff between the exactness and speed of attaining a solution. We present the effectiveness of the developed integrated approaches, Sampling Based Progressive Hedging Algorithm (SBPHA) and Discarding SBPHA (d-SBPHA), over the pure strategies (i.e. SAA). The validation of the methods is demonstrated through two-stage stochastic CRFLP. Promising results are attained for CRFLP, and the method has great potential to be generalized for SP problems.

Dynamic Price Dispersion of Seasonal Goods in Bertrand–Edgeworth Competition

The primary distinguishing characteristics of various revenue management systems include fixed capacities, uniform products, and customers' sensitivity to pricing when making purchase decisions. Even with only two or three competitors, the competition remains intense. This study examines the sub-game perfect Nash equilibrium of price competition within an oligopoly market where goods are perishable assets. Each seller possesses a single unit of a non-replenishable good and competes by setting prices to sell their inventory over a finite sales period. Each period, buyers seek to purchase one unit of the good, and the number of buyers entering is random. All sellers' prices are visible to buyers, and there is no cost associated with searching for goods. Through stochastic dynamic programming methods, sellers' optimal responses can be determined by treating the competition as a one-time price-setting game, considering the remaining sales periods and the current demand pattern. Assuming a binary demand model, it is demonstrated that the duopoly model exhibits a unique Nash equilibrium. In addition, it is shown that the oligopoly model does not result in price dispersion based on a discussed metric. Moreover, when a generalized demand model is considered, the duopoly model features a unique Nash equilibrium with mixed strategies, whereas the oligopoly model exhibits a unique symmetric Nash equilibrium with mixed strategies.

Liability-driven investment for pension funds: stochastic optimization with real assets

Using a multi-stage stochastic programming method, we suggest an optimal liability-driven investment (LDI) strategy for a closed defined-benefit pension fund including real assets. The objective is to jointly optimize contribution, funding ratio, and buyout cost, subject to a constraint on downside risk in terms of expected shortfall of assets relative to liabilities. Over a 10-year planning horizon, the optimal LDI strategy with a key-rate duration-matching bond portfolio outperforms the corresponding strategy with a duration-convexity matching bond portfolio as well as a strategy with an aggregate bond index-tracking portfolio. When real assets are introduced, the optimal LDI strategy includes significant investment in infrastructure and real estate, illiquidity notwithstanding. Nevertheless, delays in sales of real assets induced by illiquidity can increase downside risk.

Integrated management of abatement technology investment and resource extraction

As the need for increased renewable energy generation to meet climate goals grows, so does the demand for critical minerals in the industrial sector. This raises concerns about the availability of supply and the reliability of investment in critical minerals. Based on this, the study proposes an analytical framework for the integrated management of mineral extraction and renewable energy investments. Different from the previous strategy of separate management, this study has established an integrated model that utilizes new energy technology as the downstream application of emission reduction technology. It also utilizes copper, zinc, aluminum, rare earth elements, and other materials as the upstream raw material supply. In terms of minimizing costs, the entire supply chain process is optimized to achieve an empirical case. In addition, considering the significant uncertainty in the process of reducing emissions from new energy technologies, this study also incorporated a forward control stochastic programming method to model and solve it. We obtained confidence intervals for optimal technological investment paths and metal mining paths under various scenarios. Our findings indicate that: (1) Biomass, as a renewable energy technology with the lowest resource demand, should be prioritized for development. However, there is significant uncertainty surrounding the installed capacity of photovoltaics. (2) Copper ore reserves have little impact on the development of renewable energy technologies, while zinc ore and aluminum ore have a greater impact on wind energy and photovoltaics, respectively. (3) By optimizing the cut-off ratio of raw ore under a specific amount of ore storage, we can ensure the availability of materials needed for downstream energy technology.

Problem-based scenario generation by decomposing output distributions

Scenario generation is required for most applications of stochastic programming to evaluate the expected effect of decisions made under uncertainty. We propose a novel and effective problem-based scenario generation method for two-stage stochastic programming that is agnostic to the specific stochastic program and kind of distribution. Our contribution lies in studying how an output distribution may change across decisions and exploit this for scenario generation. From a collection of output distributions, we find a few components that largely compose these, and such components are used directly for scenario generation. Computationally, the procedure relies on evaluating the recourse function over a large discrete distribution across a set of candidate decisions, while the scenario set itself is found using standard and efficient linear algebra algorithms that scale well. The method’s effectiveness is demonstrated on four case study problems from typical applications of stochastic programming to show it is more effective than its distribution-based alternatives. Due to its generality, the method is especially well suited to address scenario generation for distributions that are particularly challenging.

DYNAMIC PROGRAMMING FOR SOLVE THE INVENTORY MANAGEMENT PROBLEM IN LOGISTICS

Currently, there is a problem of methods insufficient efficiency for finding solutions to the inventory management problem. The research object is the process of solving inventory management problems. The research subject is methods for finding a solution to the inventory management problem based on dynamic programming. The research goal is to increase the efficiency of finding a solution to the inventory management problem through dynamic programming. A method based on deterministic dynamic programming, a method based on stochastic dynamic programming, a method based on Q-learning, and a method based on SARSA were applicated for the inventory management problem. There are advantages of the methods. of Methods modification of deterministic and stochastic dynamic programming, Q-learning, and SARSA due to dynamic parameters makes it possible to increase the learning speed while maintaining the root-mean-square error of the method. The numerical study made it possible to evaluate the methods (for modifying the deterministic and stochastic dynamic programming methods, the number of iterations is close to the number of stages; for both methods of deterministic and stochastic dynamic programming, the root mean square error was 0.02; for modifying the Q-learning and SARSA methods, the number of iterations was 300, for both methods of Q-learning and SARSA, the root mean square error was 0.05). These methods make it possible to expand the scope of dynamic programming, which is confirmed by their adaptation to the inventory management problem and helps to increase the intelligent computer systems efficiency for general and special purposes. The application of these methods for a wide class of artificial intelligence problems are the prospects for further research.

Reinsurance, investment and the rationality with a diffusion model approximating a jump model

This article investigates the optimal excess-loss reinsurance, investment, and the rationality of using a diffusion model to approximate a jump model. We assume that the instantaneous rate of return of the risky asset is modelled by an Ornstein-Uhlenbeck (O-U) process, and the insurance claims are modeled by a compound Poisson (CP) process and a diffusion approximation (DA) model. By using the stochastic dynamic programming method, the closed-form expressions for the optimal reinsurance and investment strategies and the corresponding value functions are derived. We find that the insurer with the CP claim model always has higher reinsurance demand than the insurer with the DA claim model. Moreover, the error of using the DA model to approximate the CP model is very much dependent on the reinsurance price. Numerical analysis shows that the insurer adjusts the investment strategy as the state of the financial market changes.

Risk-averse bi-level planning model for maximizing renewable energy hosting capacity via empowering seasonal hydrogen storage

Renewables (i.e., solar and wind power) in the Nordic area have highly seasonal characteristics, which severely restrict the wide development of renewables since they can be excessive in summer but insufficient in winter with intermittent outputs. The conventional battery energy storage system (BESS) with short-term adjustment functionality cannot eliminate the seasonal imbalance of renewables. In this regard, a risk-based bi-level planning model is presented to maximize the hosting capacity (HC) of renewables through configuring seasonal hydrogen storage (SHS) and BESS. Specifically, the upper level applies a multi-objective scheme to maximize HC and minimize the investment cost simultaneously. In turn, the lower level is driven by maximizing the profits of the distribution system operator (DSO) with the implementation of price-based demand response (PBDR). Due to forecasting errors of renewables and load resulting from intermittent output and random behaviors, a stochastic programming (SP) method is developed to address and adapt multiple uncertain fluctuations. Moreover, the conditional value-at-risk (CVaR) is introduced to measure the effect of the risks raised by multiple uncertainties on system operation. Finally, numerical studies are employed to verify the effectiveness of the proposed model. Compared to the cases without considering PBDR and SHS, the total renewable energy HC in the proposed model is increased by 2.30 MW and 0.37 MW, respectively, while the yearly cost benefit of DSO is enhanced by 28063.2 US$ and 17823.7 US$, respectively. The promising results demonstrate the empowerment of SHS can promote the cross-seasonal consumption of renewables, effectively maximizing the HC and improving the economic operation of distribution systems.

Multi-objective migrating birds optimization for solving stochastic home health care routing and scheduling problems considering caregiver working time constraints

Currently, many countries are suffering from a heavy burden caused by population aging since the elderly occupy a large number of medical resources. Home health care (HHC) is a prospective approach to relieve and cope with this issue. This study addresses a multi-objective HHC routing and scheduling problem (HHCRSP) considering uncertainties and caregivers’ working time balance. First, a stochastic programming method is employed to formulate the uncertain service time and travel time, and then a chance-constrained programming model is developed to minimize the total operation cost and penalty cost of violating allowable time windows. Second, a multi-objective migrating birds optimization incorporating a stochastic simulation method is proposed. An ensemble of population iteration and external archive search is designed to obtain high-quality solutions. A stochastic simulation strategy is devised to evaluate the feasibility of obtained solutions and calculate their objective values. At last, numerical experiments are implemented for solving an extending benchmark set. The comparisons and discussions verify the better performance of the proposed algorithm compared with its peers.

Improving efficiency and sustainability of water-agriculture-energy nexus in a transboundary river basin under climate change: A double-sided stochastic factional optimization method

In this study, a double-sided stochastic fractional programming (DSFP) method is developed for identifying optimal water-allocation schemes of water-agriculture-energy nexus (WAEN) system under considering climate change impact. The advantages of DSFP are (i) handling complex uncertainty of double-sided randomness, (ii) addressing conflicting objectives with optimal system efficiency, and (iii) achieving trade-off between marginal benefit and system risk. Then, a DSFP-based water-agriculture-energy nexus (DSFP-WAEN) model is formulated for water resources allocation in a transboundary river basin of Central Asia, where 48 scenarios are designed to examine the impacts of climate change, irrigation efficiency, and system risk over a long-term planning horizon (2026–2050). Results reveal that: (i) among all agricultural activities, cash crop cultivation shows the greatest change in adaption to climate change, with the share of cash crop cultivation increasing to 70.7% by 2050 under RCP4.5; (ii) Kazakhstan, which has the largest irrigation needs and is the county most sensitive to water supply risk, should be the first to be constrained in the case of severe water shortages and poor delivery levels; (iii) the improvement of irrigation efficiency can increase the inflow to the Aral Sea by 2.7%, and reduce water loss of infield irrigation by 11.3% even under severe water shortage (i.e. p = 0.01). Compared with the models based on the traditional stochastic programming and single-objective methods, DSFP-WAEN has advantages in optimizing water-use efficiency and reflecting system complexity with flexible solutions. To coordinate transboundary water conflicts, strategies from both demand and supply sides related to quota management and water-saving techniques are suggested for the Syr Darya River basin, such as clean energy transition, drip irrigation promotion and canal system improvement. From a long-term planning perspective, WAEN schemes should be adapted to risk attitude and climate change, which can help address water scarcity and achieve future sustainability.

A Stochastic Augmented Lagrangian Method for Stochastic Convex Programming

A Stochastic Augmented Lagrangian Method for Stochastic Convex Programming

Entropy-metric estimation of the small data models with stochastic parameters

The formalization of dependencies between datasets, taking into account specific hypotheses about data properties, is a constantly relevant task, which is especially acute when it comes to small data. The aim of the study is to formalize the procedure for calculating optimal estimates of probability density functions of parameters of linear and nonlinear dynamic and static small data models, created taking into account specific hypotheses regarding the properties of the studied object. The research methodology includes probability theory and mathematical statistics, information theory, evaluation theory, and stochastic mathematical programming methods. The mathematical apparatus presented in the article is based on the principle of maximization of information entropy on sets determined as a result of a small number of censored measurements of “input" and “output" entities in the presence of noise. These data structures became the basis for the formalization of linear and nonlinear dynamic and static models of small data with stochastic parameters, which include both controlled and noise-oriented input and output measurement entities. For all variants of the above-mentioned small data models, the tasks of determining the optimal estimates of the probability density functions of the parameters were carried out. Formulated optimization problems are reduced to the forms canonical for the stochastic linear programming problem with probabilistic constraints.

Risk-averse stochastic capacity planning and P2P trading collaborative optimization for multi-energy microgrids considering carbon emission limitations: An asymmetric Nash bargaining approach

The increasing penetration of renewable energy with stochastic characteristics and continuous refinement of carbon emission policies put forward higher requirements for the construction of new power systems. This paper proposes a risk-averse stochastic capacity planning and peer-to-peer (P2P) trading collaborative optimization method for multi-energy microgrids (MEMGs) considering carbon emission limitations. First, a cooperative operation model for MEMGs considering the capacity planning of distributed generation units, uncertainty from renewable energy generations, carbon emission limitations and P2P electricity trading among MEMGs is formulated. Second, a risk-averse stochastic programming method is applied to avoid the potential risk losses caused by randomness and intermittence of renewable energy. Third, an asymmetric Nash bargaining approach is adopted to ensure the fair allocation of benefits and maintain the willingness of individual MEMGs to participate in cooperation. Then, to protect the privacy of individual MEMGs belonging to different stakeholders, the alternating direction method of multipliers is used to solve the two subproblems in a distributed manner. Meanwhile, to further alleviate the computation burdens, a diagonal quadratic approximation method is applied to linearize the quadratic penalty term in the augmented Lagrangian function and realize the parallel solution of all optimization subproblems. Moreover, the influence of different carbon emission targets on the optimal resource combination strategy of the system is investigated by introducing carbon emission factors. Simulations on different models, strategies, and distributed algorithms are conducted to verify the effectiveness and superiority of the proposed method.

Ensemble Variance Reduction Methods for Stochastic Mixed-Integer Programming and their Application to the Stochastic Facility Location Problem

Sample average approximation (SAA), the standard approach to stochastic mixed-integer programming, does not provide guidance for cases with limited computational budgets. In such settings, variance reduction is critical in identifying good decisions. This paper explores two closely related ensemble methods to determine effective decisions with a probabilistic guarantee. (a) The first approach recommends a decision by coordinating aggregation in the space of decisions as well as aggregation of objective values. This combination of aggregation methods generalizes the bagging method and the “compromise decision” of stochastic linear programming. Combining these concepts, we propose a stopping rule that provides an upper bound on the probability of early termination. (b) The second approach applies efficient computational budget allocation for objective function evaluation and contributes to identifying the best solution with a predicted lower bound on the probability of correct selection. It also reduces the variance of the upper bound estimate at optimality. Furthermore, it adaptively selects the evaluation sample size. Both approaches provide approximately optimal solutions even in cases with a huge number of scenarios, especially when scenarios are generated by using oracles/simulators. Finally, we demonstrate the effectiveness of these methods via extensive computational results for “megascale” (extremely large scale) stochastic facility location problems. History: Accepted by Pascal Van Hentenryck, Area Editor for Computational Modeling: Methods & Analysis. Funding: This work was supported by The Office of Naval Research [Grant N00014-20-1-2077] and the Air Force Office of Scientific Research [Grant FA9550-20-1-0006]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2021.0324 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2021.0324 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

A novel day-ahead scheduling model to unlock hydropower flexibility limited by vibration zones in hydropower-variable renewable energy hybrid system

Giving full play to the flexibility of hydropower and integrating more variable renewable energy (VRE) is of great significance for accelerating the transformation of China's power energy system. For middle- to large-sized hydropower units, its irregular vibration zone (VZ) is a major factor affecting its flexibility, because VZs limits the power output and adjustment range. In the existing studies on hydropower unit commitment and flexibility scheduling, VZs only limits the output of hydropower units, while the impact on the flexibility adjustment ability of units is ignored. The hydropower flexibility in day-ahead scheme is overestimated, which increases the risk of system flexibility shortage causing power curtailment and load loss. In this study, a novel day-ahead scheduling model considering the flexibility limited by the VZs and the probability of flexibility shortage is constructed for the day-ahead generation scheme of hydropower-VRE hybrid generation system (HVHGS) to optimize the target load, VRE output and hydropower unit commitments. The impaction of the VZs on hydropower flexibility is firstly modeled by chance constraints and stochastic programming method. Moreover, a data-driven model based on machine learning and an efficient solving approach based on successive linear programming is carry out to describe the uncertainty of VRE output more realistically and ensure the timeliness of optimization scheme, respectively. The proposed model is applied to a real hydropower station in the Hongshui River Basin in China. In 16 representative scenarios, the proposed model can complete the optimization in an acceptable time, with a maximum of 444.57 s. Compared with the traditional interval optimization model, the proposed model effectively improves the flexibility supply capacity of hydropower in the day-ahead scheme. The maximum reduction value of flexibility shortage probability and expectation reach 98.77% and 442.66 MW, respectively. In particular, the flexibility of the model is most obvious under heavy load demand in flood season, and it is almost not at the cost of daily target load adjustment, which provides practical reference for decision makers.

A Generalized Control Paradigm for Storage Systems: Optimal Energy Management and Stability Certificates

In this brief, we propose a suboptimal control update for lossless storage systems that operates based on the instantaneous value of the load power and generalizes previously suggested solutions for energy management problems. The proposed control update performs under uncertain conditions and does not require statistical information about the load profile. We leverage tools from Pontryagin’s minimum principle and stochastic dynamic programming methods to derive the control strategy and then investigate its stability properties using the Lyapunov analysis. However, the analysis of this control paradigm presents significant challenges, among which the key difficulty arises due to the presence of nonlinear feedback terms. To address this challenge, we first introduce a coordinate transformation, which transforms the original problem into an equivalent stabilization problem, and then verify its stability properties using the Lyapunov function theory and LaSalle’s invariance principle. We show that the proposed control policy works for both continuous-time and stochastic load profiles, and it keeps the stored energy within the capacity bounds by following the optimal path.

Unit Commitment Problem with Energy Storage Under Correlated Renewables Uncertainty

Unit Commitment Problem with Energy Storage Under Correlated Renewables Uncertainty” introduces a novel approach to address the challenges of renewable integration. The study acknowledges the growing variability and correlation in power availability due to renewable generation and proposes a day-ahead unit commitment (UC) problem formulation that incorporates energy storage and considers multistage correlated uncertainty. Using a variant of the stochastic dual dynamic programming (SDDP) method, which can handle temporal correlations effectively, the researchers solve the complex UC problem. Results obtained from the IEEE 118-bus system demonstrate the significant advantages of considering multistage uncertainty and correlations. Applying their approach to the Chilean power system, the researchers present superior UC solutions that adapt generation to changing uncertainty at a lower cost. Additionally, they propose a more efficient deterministic UC solution that outperforms current industry practices. These advancements promise to enhance the integration of renewable energy sources, enabling a more sustainable and environmentally friendly future.

Buffer Times Between Scheduled Events in Resource Assignment Problem: A Conflict-Robust Perspective

Problem definition: In many resource scheduling problems for services with scheduled starting and completion times (e.g., airport gate assignment), a common approach is to maintain appropriate buffer between successive services assigned to a common resource. With a large buffer, the chances of a “crossing” (i.e., a flight arriving later than the succeeding one at the gate) will be significantly reduced. This approach is often preferred over more sophisticated stochastic mixed-integer programming methods that track the arrival of all the flights to infer the number of “conflicts” (i.e., a flight arriving at a time when the assigned gate becomes unavailable). We provide a theoretical explanation, from the perspective of robust optimization for the good performance of the buffering approach in minimizing not only the number of crossings but also the number of conflicts in the operations. Methodology/results: We show that the buffering method inherently minimizes the worst-case number of “conflicts” under both robust and distributionally robust optimization models using down-monotone uncertainty sets. Interestingly, under down-monotone properties, the worst-case number of crossings is identical to the worst-case number of conflicts. Using this equivalence, we demonstrate how feature information from flight and historical delay information can be used to enhance the effectiveness of the buffering method. Managerial implications: The paper provides the first theoretical justification on the use of buffering method to control for the number of conflicts in resource assignment problem. Funding: This work was supported by the 2019 Academic Research Fund Tier 3 of the Ministry of Education-Singapore [Grant MOE-2019-T3-1-010] and the Research Grants Council of Hong Kong SAR, China [Grant PolyU 152240/17E]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/msom.2022.0572 .